Path-Bicolorable Graphs

  • Authors:

  • Andreas Brandstädt;Martin C. Golumbic;Van Bang Le;Marina Lipshteyn

  • Affiliations:

  • Institut für Informatik, Universität Rostock, Rostock, Germany D-18051;Caesarea Rothschild Institute and Department of Computer Science, University of Haifa, Haifa, Israel;Institut für Informatik, Universität Rostock, Rostock, Germany D-18051;Caesarea Rothschild Institute and Department of Computer Science, University of Haifa, Haifa, Israel

  • Venue:
  • Graph Theory, Computational Intelligence and Thought
  • Year:
  • 2009

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Abstract

In this paper, we introduce the notion of path-bicolorability that generalizes bipartite graphs in a natural way: For k *** 2, a graph G = (V ,E ) is P k -bicolorable if its vertex set V can be partitioned into two subsets (i.e., colors) V 1 and V 2 such that for every induced P k (i.e., path with exactly k *** 1 edges and k vertices) in G , the two colors alternate along the P k , i.e., no two consecutive vertices of the P k belong to the same color V i , i = 1,2. Obviously, a graph is bipartite if and only if is P 2 -bicolorable, every graph is P k -bicolorable for some k and if G is P k -bicolorable then it is P k + 1 -bicolorable. The notion of P k -bicolorable graphs is motivated by a similar notion of cycle-bicolorable graphs introduced in connection with chordal probe graphs. Moreover, P 3 - and P 4 -bicolorable graphs are closely related to various other concepts such as 2-subcolorable graphs, P 4 -bipartite graphs and alternately orientable graphs. We give a structural characterization of P 3 -bicolorable graphs which also implies linear time recognition of these graphs. Moreover, we give a characterization of P 4 -bicolorable graphs in terms of forbidden subgraphs.